As with all questions of definite vs. indefinite article usage, there are slight nuances, depending on the sentence, in which one might be preferred to the other. Let's look at some examples.
- Caroline is typing her report on the computer.
- David sits in front of a computer at work all day, so he likes to get outside in the evenings.
In these two examples, it is implied that there is a specific computer (first example) and not (second example). However, both sentences could be grammatically correct with either article.
In your example, I think it would simply work better to say "in front of a/the computer", and neither article sounds particularly better to me. If you do a Google news search for "at a computer" and "at the computer", you'll see that the verbs preceding the prepositional phrase are often "look" or "stare".